Bridges 2010: Mathematics, Music, Art, Architecture, Culture About Weaving and Helical Holes Rinus Roelofs Sculptor Lansinkweg 28 7553AL Hengelo The Netherlands E-mail: [email protected] www.rinusroelofs.nl Abstract Weaving is an invention of man, and maybe one of the most important inventions in the field of construction. There are many weave patterns but most weave patterns are based on just two different grids, the plain weave and the three way weave. The development of weaving is mainly connected to the possibilities of manufacturing. In this paper I want to investigate this ‘passing one thread over another’-concept from a mathematical-artistic point of view. 1. Introduction 1.1. Over and Under. In Leonardo’s painting on the ceiling of the Sala delle Asse in the Sforza Castle in Milan (Figure 1) we can see different weaving patterns: the knotted ropes and the interwoven branches. Both can only be realized with the aid of man. Nature doesn’t weave by itself, at least not in the way Leonardo shows us in his painting. When we look for a definition of weaving we can find the following: weaving is the textile art in which two distinct sets of yarns or threads, called the warp and the filling or weft, are interlaced with each other to form a fabric or cloth. To define the concept of weaving in a more general way we could say that a weaving is a line pattern in which for each pair of adjacent crossing points on the line(s) the position of that line changes from ‘over’ to ‘under’ or from ‘under’ to ‘over’ (Figure 2a,b). Figure 1: Leonardo da Vinci Figure 2a: Lines Figure 2b: Weaving 1.2. Graphs and Tilings. Lines in a weave pattern can be lines with a start and an endpoint (Figure 2b) or closed loops (Figure 3). And a weave pattern can even be made with one single line (Figure 4a). When we project a weave pattern on the plane the resulting figure can be seen as a graph in which each vertex represents a crossing point of the weave pattern. The degree of each vertex is four, because of the two lines in the weaving pattern crossing each other. And therefore the projected pattern can be transformed into a tiling which can be colored with just two colors in such a way that adjacent tiles always have different colors (Figure 4). This works both ways, so we can construct a weave pattern from any tiling that can be colored in this way and has only vertices with degree four. 75 Roelofs Figure 3: Closed loops. Figure 4a: Weaving. Figure 4b: Tiling. 2. Weave Patterns 2.1. Tilings and Weave Patterns. When we look at the Archimedean tilings [1] we see that there are four tilings which can be colored in the way described above. Two of them, (4.4.4.4) and (3.6.3.6) are the underlying patterns of most of the weavings used in fabrics and baskets. The weaving derived from (4.4.4.4) is known as the plain weave (Figure 5), and the other as the three way weave (Figure 6). Figure 5a: Tiling 4.4.4.4. Figure 5b: Plain weave. Figure 6a: Tiling 3.6.3.6. Figure 6b: Three way weave. 2.2. Closed Loops. The third Archimedean tiling that can be colored like a checkerboard is (3.4.6.4) (Figure 7a). The weaving derived from this tiling is that it is a weaving with closed loops (Figure 7b). Figure 7a: Tiling 3.4.6.4 Figure 7b: Weaving with closed loops. The last one, tiling (3.3.3.3.3.3), can be colored in the right way but can not be transformed into a weaving pattern because the degree of each vertex is not four but six. 3. Transforming Patterns into Two-Color Patterns 3.1. Connecting intersection points. Each tiling can be used to create a new tiling that can be colored as a checkerboard. When we add the dual pattern to an existing pattern we will get intersection points between both patterns. Each of those points lays on a line which is the edge of two adjacent tiles. On both tiles this edge is connected to exactly two other edges, on which there are also intersection points. So we have four 76 About Weaving and Helical Holes direct neighbour points for each intersection point. Connecting each intersection point to the four direct neighbour points will thus give us a graph in which each vertex has degree four. So we will get a tiling which can be colored as a checkerboard, that can be used as an underlying pattern for a weaving. In Figure 9 you can see the result of this process applied on the Archimedean tiling (3.3.4.3.4) (Figure 8). This operation corresponds to the “ambo” operation that is part of Conway Polyhedron notation [2]. Figure 8a: Tiling. Figure 8b: Transformed tiling. Figure 9: Weave. 3.2. The other Archimedean Tilings. As can be seen in Figure 10 to 13 we can derive new weave patterns from the other Archimedean tilings. Note that when we apply this process on tiling (3.3.3.3.3.3) or on tiling (6.6.6) in both cases we will get tiling (3.6.3.6), which already explained in Section 2.1. Starting with the tiling (4.4.4.4) brings us back to tiling (4.4.4.4) again. Figure 10: (6.3.3.3.3). Figure 11: (3.12.12). Figure 12: (4.8.8). Figure 13: (4.6.12). 4. Weaving Patterns on Polyhedra 4.1. Rings on the Sphere. In the group of regular and semi regular polyhedra there are, except from the anti prisms, five polyhedra that can be colored in a checkerboard fashion. And they all have vertices with degree 4. This gives us five weave patterns on the sphere as can be seen in Figure 14. In Japanese Temari Balls and also in Alan Holden’s Orderly Tangles [3] we can find many examples of the use of these basic patterns. But with the procedure described in section 3.1 we can also derive weave patterns from the other Platonic and Archimedean solids. In Figures 15 to 20 some examples are shown. In the weavings we still get closed loops but these lines are not laying in a cutting plane of the polyhedron as in the first five. And in some cases (snub cube and snub dodecahedron), the loops go around twice, crossing themselves a few times. The weavings shown in Figures 15 – 20 are based on the Rhombic Truncated Cuboctahedron, the Truncated Dodecahedron, the Rhombic Truncated Icosidodecahedron, the Snubcube and the Snubdodecahedron. 77 Roelofs Figure 14: Five spherical weaves. Figure 15: R.T.Cuboc. Figure 16: T.Icos. Figure 17: T.Dodec.. Figure 18: R.T.Icosidodec. Figure 19: Snub Cube Figure 20: Snub Dodec. 5. Twist 5.1. Transformations. As tilings can be used to generate weaving patterns, also weaving patterns themselves can be used to generate other weaving patterns. We will discuss and illustrate some of the possible transformations that can be used to generate new weaving patterns and also new weaving structures. 5.2. From lines to closed loops. In a plain weave there are straight crossings between the lines. You can say that each line at each crossing point takes the second exit. If we change that, so instead of the taking the second exit we take the third and we do that with both lines of the crossing, we will get a twist at the crossing point of our weaving. The first twist (Figure 21a) will just cause a defect in the weaving. But applying this twist on every crossing point in the weaving will transform our line weaving into a weaving with closed loops (Figure 21c). Figure 22 shows the result of this transformation starting with the three way weave. Figure 21a: One defect Figure 21b: One loop. Figure 21c: Loop weaving. Figure 22: Three way. 5.3. Change of Rhythm. The next operation can be seen as a change of rhythm: instead of over-under- over-under (notation: 1-1) we will change to over-over-under-under (notation: 2-2). To achieve this we just change some of the crossings from over-under to under-over. In Figure 23 you can see the result of this operation applied on the weaving of Figure 21c. 78 About Weaving and Helical Holes Figure 23: Rhythm 2-2. Figure 24a: Rhythm 3-3. Figure 24b: Rhythm 4-4. Figure 24c: Surface. The rhythm of the weaving with the ellipses in Figure 24a is 3-3 and when we make an extra twist in each of the ellipses we will get the weaving of Figure 24b in which the rhythm is 4-4. 5.4. Making Surfaces. It appeared to be possible to create a surface on the edges of the weaving of Figure 24b like a soap-film. Whether this is due to the fact that the rhythm is even-even (4-4) or if this is possible with all even-even loop weavings still has to be examined. The resulting surface however is a self interwoven surface and as such a new weaving structure derived from a weaving pattern. In Figures 25 to 27 a few more examples are shown of self interwoven surfaces created from twisted loop weavings. The starting point for Figure 25 is the three way weave. In Figure 26 the Archimedean tiling (3.4.6.4) can be recognized.